Nice find. I found another recording here.
https://www.youtube.com/watch?v=uQcgdSQ ... 3w&index=3. It’s a medley of three songs, with the first one being Ím Bím Babaró.
And I found some lyrics for Ím Bím Babaró here.
https://www.duchas.ie/en/cbe/9000685/71 ... entScript=There are 4 lines per verse, with the first and last lines matching within each verse, and also between verses. So only the second and third lines change across the verses. The lyrics are whimsical in nature, such that they would appeal to a small child.
However, the lyrics generally differ between the three sources (apart from the first and last lines of each verse, which all three sources have in common). The recording you found has 5 verses. The recording I found has 4 verses, with the 3rd and 4th verses being in English (the English verses are not translations of the first two verses, which are in Irish). The third source, which only gives lyrics, has three verses. The only match in lyrics across the three sources is between those of the second verse of the recording you found and the second verse of the recording I found.
Your mother’s lyrics bear a slight similarity to the source that only gives the lyrics. Specifically, the second line of the first verse has “Táim ag fanacht le mo bhuachaillín” (meaning: I am waiting for my little boy). This seems similar to, but not the same as, what I had labelled as Lines 2 and 3 of your mother’s recording (although the first three words are identical). Maybe she’s trying to say “mo bhuachaillí” (my boys) here instead of ‘mo bhuachaillín” (my little boy).
Anyway, It’s a very pleasant melody. I'm sure your mother will enjoy hearing it.